I am supposed to find some simple and basic examples for semimartingales $X$ and predictable processes $H$ such that $H\notin L(X)$, that means $H$ is not integrable with respect to $X$.
Do you have ...

We have a Gaussian process $X$, $X_t:=B_t - tB_1$, where $B$ is a $BM$, $t\in[0,1]$.
Let $\nu$ be the law of $X$ and $\mu$ the Wiener measure.
How can I show that $\mu$ is singular with respect to $...

If I start with the definition of a predictable process as a measurable mappings on the predictable $\sigma$-algebra generated by sets like
$$
(s,t]\times F, \quad s<t, \quad F \in \mathcal{F}_s
$$
...

Suppose we have an SDE of the form $dX=Ndt+MdB$. Ito lemma lets us know that F(X) also is an Ito process. This would imply in particular that the term $f^{'}MdB$ is a martingale.
I wounder if this is ...

Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$
$M$ be a real-valued continuous local $\mathcal F$-martingale on $(\...

I'm searching results of the following form: Let $G_n,G$ be generators of related Markov processes $X^{(n)},X$ on a common state space $S$. Assume there is a collection of sets $F_n\subseteq S$ such ...

I am currently reading the famous almost sure blog by George Lowther about stochastic calculus. I am currently reading the section about the Burkholder-Davis-Gundi-Inequality (BDG inequality). At the ...

Let $N_1(t)$ be a delayed renewal process and $N_2(t)$ be an ordinary renewal process such that $N_1(t)\geq_{st}N_2(t)$. Consider a renewal process $Z(t)$ with the same inter-arrivall distribution as $...

Let $X$ be a stochastic process and $T$ a stopping time. Then one forms the random variable $X_T$.
I have a quite vague question: apparently $X_T$ will be quite different if we replace $X$ by one of ...

Let $B$ be a BM, $t\in (0,1)$. Calculate $E(B_t|B_1)$.
There is a hint: If we have a sequence of i.i.d. variables $(X_i)_{i\in \mathbb{N}}$ and the first moment of $X_1$ exists, then for $S_n:=\sum_{...

I'm currently stuck in a stochastic approximation problem in my research for a while.
Suppose I'm approximating $(a_n, b_n, c_n)'$ using stochastic recursive sequence. I'm not interested in whether $...

I was wondering how to prove/compute the differential of the product of two Brownian motions. I know how to do it in case they are independent as follows:
Suppose $dX_t= \mu_t dt +\sigma_t dW_t$ and ...

Assume I have a collection of real-valued measurable functions $(X_i)_{i \in I}$ on the measurable space $(\Omega,\mathcal{F})$. Let $N:\Omega \rightarrow 2^\Omega$ such that for every $\omega \in \...

Assume $X=(X_t)_{t \in [0,\infty)}$ is a RCLL adapted process. I set $X_{-t}:=\lim_{s \rightarrow t, s < t}X_s$ for $t \in (0,\infty)$ and $X_{-0}=0$. I want to show that $(X_{-t})_{t \in [0,\infty)...

Let $(\Omega,\mathcal F,\mathcal F_t,P)$ be a probability space and $W(t)$ be a Brwonian Motion defined on it. Let $M(t)$ be a bounded martingale orthogonal to $W(t)$, i.e., there exists a constant $C&...

So I have a model defined by the following transition states
$$T_1\equiv T\left(x_1+\frac{1}{N},x_2-\frac{1}{N}|x_1,x_2\right)=rx_1x_2+\epsilon x_2$$
$$T_2\equiv T\left(x_1-\frac{1}{N},x_2+\frac{1}{N}|...